Veronese Surface Represented by an Intersection of Quadratics

I was reading Algebraic Geometry by Harris and he introduces the Veronese surface defined by

$$v : \mathbb{P}^2 \rightarrow \mathbb{P}^5$$ $$v : [X_0,X_1,X_2] \rightarrow [X_0^2,X_1^2,X_2^2, X_0X_1,X_0X_2,X_1X_2]$$

He states that this could be represented by the intersection of quadratics $X^IX^J = X^K X^L$, where $X^I,X^J,X^K,X^L$ are appropriately chosen monomials in $X_0,X_1,X_2$. I can see why it lies in this intersection but cannot see at all why it is determined exactly by this intersection.

Also, there have been other questions like this in the book and they've mostly involved a bit of algebra, but there doesn't seem to be any pattern in them and I'd be pretty at a loss of the examples weren't quadratic, but say cubic.

How can we, in general, tell when the image of a polynomial map from $\mathbb{P}^a \rightarrow \mathbb{P}^b$ like of a collection of polynomials, and actually find the set of polynomials that are necessary to describe it?

There are probably tricks one can use to solve this particular problem, but one can solve this sort of problem in general by using Gröbner bases. I highly recommend Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms: they treat this exact problem in Chapter 3 Elimination, particularly in $\S3$ Implicitization. The main result is the following theorem.

Theorem 1 (Polynomial Implicitization) If $k$ is an infinite field, let $F: k^m \to k^n$ be the function determined by the polynomial parametrization \begin{align*} x_1 &= p_1(t_1, \ldots, t_m)\\ &\ \, \vdots\\ x_n &= p_n(t_1, \ldots, t_m) \, . \end{align*} Let $I = \langle x_1 - p_1, \ldots, x_n - p_n \rangle$ and let $I_m = I \cap k[x_1, \ldots, x_n]$ be the $m^\text{th}$ elimination ideal. Then $\mathbb{V}(I_m)$ is the smallest variety in $k^n$ containing $F(k^m)$.

This deals with affine varieties, and the results in $\S$5 of Chapter 8 can be used to extend to projective varieties.

Let $z_0, \ldots, z_5$ be the coordinates on $\mathbb{P}^5$. The image of $v$ is given parametrically by the equations $$ z_0 = x_0^2 \qquad z_1 = x_1^2 \qquad z_2 = x_2^2 \qquad z_3 = x_0 x_1 \qquad z_4 = x_0 x_2 \qquad z_5 = x_1 x_2 \, . $$ Let $I = \langle z_0-x_0^2, z_1-x_1^2, z_2-x_2^2, z_3-x_0 x_1, z_4-x_0 x_2, z_5-x_1 x_2 \rangle \trianglelefteq k[x_0,x_1,x_2,z_0, \ldots, z_5]$ be the ideal corresponding to this parametrization. I computed a Gröbner basis for $I$ in Sage with respect to the graded lexicographic monomial ordering $x_0 > x_1 > x_2 > z_0 > \cdots > z_5$ as follows:

R.<x0,x1,x2,z0,z1,z2,z3,z4,z5> = PolynomialRing(QQ,9,order = "deglex")
gens = [z0-x0^2, z1-x1^2, z2-x2^2, z3-x0*x1, z4-x0*x2, z5-x1*x2]
I = Ideal(gens)
G = I.groebner_basis()

This outputs a Gröbner basis containing $20$ generators. Those only involving the $z_i$ are $$ z_0 z_1 - z_3^2, \qquad z_0z_2 - z_4^2, \qquad z_0 z_5 - z_3 z_4, \qquad z_1 z_2 - z_5^2, \qquad z_1 z_4 - z_3 z_5 $$ and these are the quadrics defining the image of $v$.